Namespaces
Variants
Actions

Difference between revisions of "Lagrange bracket"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
''Lagrange brackets, with respect to variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571302.png" />''
+
<!--
 +
l0571301.png
 +
$#A+1 = 39 n = 0
 +
$#C+1 = 39 : ~/encyclopedia/old_files/data/L057/L.0507130 Lagrange bracket,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
''Lagrange brackets, with respect to variables  $  u $
 +
and  $  v $''
  
 
A sum of the form
 
A sum of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\sum _ { i= } 1 ^ { n }
 +
\left (
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571305.png" /> are certain functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571307.png" />.
+
\frac{\partial  q _ {i} }{\partial  u }
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571309.png" /> are canonical variables and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713011.png" /> are canonical transformations, then the Lagrange bracket is an invariant of this transformation:
+
\frac{\partial  p _ {i} }{\partial  v }
 +
-
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713012.png" /></td> </tr></table>
+
\frac{\partial  q _ {i} }{\partial  v }
  
For this reason the indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713013.png" /> on the right-hand side of (*) are often omitted. The Lagrange bracket is said to be fundamental when the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713015.png" /> coincide with some pair of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713016.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713017.png" />. From them one can form three matrices:
+
\frac{\partial  p _ {i} }{\partial  u }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713018.png" /></td> </tr></table>
+
\right )  \equiv  [ u , v ] _ {p , q }  ,
 +
$$
  
the first two of which are the zero, and the last one is the unit matrix. There is a definite connection between Lagrange brackets and [[Poisson brackets|Poisson brackets]]. Namely, if the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713020.png" />, induce a diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713021.png" />, then the matrices formed from the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713023.png" /> are inverse to each other.
+
where  $  q = ( q _ {1} \dots q _ {n} ) $
 +
and  $  p = ( p _ {1} \dots p _ {n} ) $
 +
are certain functions of  $  u $
 +
and  $  v $.
 +
 
 +
If  $  q = ( q _ {1} \dots q _ {n} ) $
 +
and  $  p = ( p _ {1} \dots p _ {n} ) $
 +
are canonical variables and  $  Q = Q ( q , p ) $,
 +
$  P = P ( q , p ) $
 +
are canonical transformations, then the Lagrange bracket is an invariant of this transformation:
 +
 
 +
$$
 +
[ u , v ] _ {q , p }  = \
 +
[ u , v ] _ {Q , P }  .
 +
$$
 +
 
 +
For this reason the indices  $  q , p $
 +
on the right-hand side of (*) are often omitted. The Lagrange bracket is said to be fundamental when the variables  $  u $
 +
and  $  v $
 +
coincide with some pair of the  $  2n $
 +
variables  $  q , p $.
 +
From them one can form three matrices:
 +
 
 +
$$
 +
[ p , p ]  = \
 +
\{ [ p _ {i} , p _ {j} ] \} _ {i , j = 1 }  ^ {n} ,\ \
 +
[ q , q ] ,\  [ q , p ] ,
 +
$$
 +
 
 +
the first two of which are the zero, and the last one is the unit matrix. There is a definite connection between Lagrange brackets and [[Poisson brackets|Poisson brackets]]. Namely, if the functions $  u _ {i} = u _ {i} ( q , p ) $,  
 +
$  1 \leq  i \leq  n $,  
 +
induce a diffeomorphism $  \mathbf R  ^ {2n} \rightarrow \mathbf R  ^ {2n} $,  
 +
then the matrices formed from the elements $  [ u _ {i} , u _ {j} ] $
 +
and $  ( u _ {j} , u _ {i} ) $
 +
are inverse to each other.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Lagrange,  "Oeuvres" , '''6''' , Gauthier-Villars  (1873)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.T. Whittaker,  "Analytical dynamics of particles and rigid bodies" , Dover, reprint  (1944)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Lur'e,  "Analytical mechanics" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Goldstein,  "Classical mechanics" , Addison-Wesley  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Lagrange,  "Oeuvres" , '''6''' , Gauthier-Villars  (1873)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.T. Whittaker,  "Analytical dynamics of particles and rigid bodies" , Dover, reprint  (1944)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Lur'e,  "Analytical mechanics" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Goldstein,  "Classical mechanics" , Addison-Wesley  (1957)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713024.png" /> denotes the mapping: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713025.png" />, then the Lagrange bracket is equal to the product of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713027.png" /> with respect to the canonical symplectic form (cf. [[Symplectic manifold|Symplectic manifold]]) on the phase space. More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713028.png" /> is a symplectic form on a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713030.png" /> is a smooth mapping from a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713031.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713033.png" /> is an area form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713035.png" /> is a standard area form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713036.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713038.png" /> could be called the Lagrange brackets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713039.png" />. See [[#References|[a1]]], Chapt. 3.
+
If $  \psi $
 +
denotes the mapping: $  ( u , v) \mapsto ( q ( u , v), p ( u , v)) $,  
 +
then the Lagrange bracket is equal to the product of the vectors $  {\partial  \psi } / {\partial  u } $
 +
and $  {\partial  \psi } / {\partial  v } $
 +
with respect to the canonical symplectic form (cf. [[Symplectic manifold|Symplectic manifold]]) on the phase space. More generally, if $  \omega $
 +
is a symplectic form on a smooth manifold $  M $
 +
and $  \psi $
 +
is a smooth mapping from a surface $  S $
 +
to $  M $,  
 +
then $  \psi  ^ {*} \omega $
 +
is an area form on $  S $.  
 +
If $  ds $
 +
is a standard area form on $  S $,  
 +
then the function $  \psi  ^ {*} \omega /ds $
 +
on $  S $
 +
could be called the Lagrange brackets of $  \psi $.  
 +
See [[#References|[a1]]], Chapt. 3.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Abraham,  J.E. Marsden,  "Foundations of mechanics" , Benjamin/Cummings  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "Lectures in analytical mechanics" , MIR  (1975)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Abraham,  J.E. Marsden,  "Foundations of mechanics" , Benjamin/Cummings  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "Lectures in analytical mechanics" , MIR  (1975)  (Translated from Russian)</TD></TR></table>

Revision as of 22:15, 5 June 2020


Lagrange brackets, with respect to variables $ u $ and $ v $

A sum of the form

$$ \tag{* } \sum _ { i= } 1 ^ { n } \left ( \frac{\partial q _ {i} }{\partial u } \frac{\partial p _ {i} }{\partial v } - \frac{\partial q _ {i} }{\partial v } \frac{\partial p _ {i} }{\partial u } \right ) \equiv [ u , v ] _ {p , q } , $$

where $ q = ( q _ {1} \dots q _ {n} ) $ and $ p = ( p _ {1} \dots p _ {n} ) $ are certain functions of $ u $ and $ v $.

If $ q = ( q _ {1} \dots q _ {n} ) $ and $ p = ( p _ {1} \dots p _ {n} ) $ are canonical variables and $ Q = Q ( q , p ) $, $ P = P ( q , p ) $ are canonical transformations, then the Lagrange bracket is an invariant of this transformation:

$$ [ u , v ] _ {q , p } = \ [ u , v ] _ {Q , P } . $$

For this reason the indices $ q , p $ on the right-hand side of (*) are often omitted. The Lagrange bracket is said to be fundamental when the variables $ u $ and $ v $ coincide with some pair of the $ 2n $ variables $ q , p $. From them one can form three matrices:

$$ [ p , p ] = \ \{ [ p _ {i} , p _ {j} ] \} _ {i , j = 1 } ^ {n} ,\ \ [ q , q ] ,\ [ q , p ] , $$

the first two of which are the zero, and the last one is the unit matrix. There is a definite connection between Lagrange brackets and Poisson brackets. Namely, if the functions $ u _ {i} = u _ {i} ( q , p ) $, $ 1 \leq i \leq n $, induce a diffeomorphism $ \mathbf R ^ {2n} \rightarrow \mathbf R ^ {2n} $, then the matrices formed from the elements $ [ u _ {i} , u _ {j} ] $ and $ ( u _ {j} , u _ {i} ) $ are inverse to each other.

References

[1] J.L. Lagrange, "Oeuvres" , 6 , Gauthier-Villars (1873)
[2] E.T. Whittaker, "Analytical dynamics of particles and rigid bodies" , Dover, reprint (1944)
[3] A.I. Lur'e, "Analytical mechanics" , Moscow (1961) (In Russian)
[4] H. Goldstein, "Classical mechanics" , Addison-Wesley (1957)

Comments

If $ \psi $ denotes the mapping: $ ( u , v) \mapsto ( q ( u , v), p ( u , v)) $, then the Lagrange bracket is equal to the product of the vectors $ {\partial \psi } / {\partial u } $ and $ {\partial \psi } / {\partial v } $ with respect to the canonical symplectic form (cf. Symplectic manifold) on the phase space. More generally, if $ \omega $ is a symplectic form on a smooth manifold $ M $ and $ \psi $ is a smooth mapping from a surface $ S $ to $ M $, then $ \psi ^ {*} \omega $ is an area form on $ S $. If $ ds $ is a standard area form on $ S $, then the function $ \psi ^ {*} \omega /ds $ on $ S $ could be called the Lagrange brackets of $ \psi $. See [a1], Chapt. 3.

References

[a1] R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin/Cummings (1978)
[a2] F.R. [F.R. Gantmakher] Gantmacher, "Lectures in analytical mechanics" , MIR (1975) (Translated from Russian)
How to Cite This Entry:
Lagrange bracket. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lagrange_bracket&oldid=47553
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article